A175576 -id:A175576 - cp's OEIS frontend

A010466 Decimal expansion of square root of 8.

2, 8, 2, 8, 4, 2, 7, 1, 2, 4, 7, 4, 6, 1, 9, 0, 0, 9, 7, 6, 0, 3, 3, 7, 7, 4, 4, 8, 4, 1, 9, 3, 9, 6, 1, 5, 7, 1, 3, 9, 3, 4, 3, 7, 5, 0, 7, 5, 3, 8, 9, 6, 1, 4, 6, 3, 5, 3, 3, 5, 9, 4, 7, 5, 9, 8, 1, 4, 6, 4, 9, 5, 6, 9, 2, 4, 2, 1, 4, 0, 7, 7, 7, 0, 0, 7, 7, 5, 0, 6, 8, 6, 5, 5, 2, 8, 3, 1, 4, 5, 4, 7

Offset: 1

N. J. A. Sloane

Sqrt(8) = 2*sqrt(2) is the length of the longest (rigid) ladder that can be carried horizontally around a right angled corner in a hallway of unit width. - Lekraj Beedassy, Apr 19 2006
Continued fraction expansion is 2 followed by {1, 4} repeated. - Harry J. Smith, Jun 05 2009
This is the second Lagrange number. - Alonso del Arte, Dec 06 2011
Also 2*sqrt(2) is the ratio of the perimeter of a square to its diameter (diagonal length). - Rick L. Shepherd, Dec 29 2016
Uchiyama shows that every interval (n, n + c*n^(1/4)) contains an integer that is the sum of two squares, where c = 2^(3/2). - Michel Marcus, Jan 03 2018
This is the area of the eighth-smallest triangle with integer side lengths (2, 3, 3), or the seventh-smallest triangle if two smaller triangles with the same area are counted only once (see A331251). - Hugo Pfoertner, Feb 12 2020
Diameter of a sphere whose surface area equals 8*Pi. More generally, the square root of x is also the diameter of a sphere whose surface area equals x*Pi. - Omar E. Pol, Feb 13 2020
Sqrt(8) = area between the curves y = sin(x) and y = cos(x) for Pi/4 < x < 5 Pi/4; this is one of infinitely many congruent convex regions bounded by the two curves. - Clark Kimberling, May 03 2020
Area of the regular 8-gon with circumradius =1. - R. J. Mathar, Aug 24 2023

			2.828427124746190097603377448419396157139343750753896146353359475981464...
Sqrt(8) = sqrt(1+2*i*sqrt(2)) + sqrt(1-2*i*sqrt(2)) = sqrt(1/2+2*i*sqrt(3)) + sqrt(1/2-2*i*sqrt(3)), where i=sqrt(-1). - _Bruno Berselli_, Nov 20 2012
1 + 3/4 + 3*5/(4*8) + 3*5*7/(4*8*12) + 3*5*7*9/(4*8*12*16) + ... - _Bruno Berselli_, Mar 16 2014

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 187.
S. R. Finch, Moving Sofa Constant, Sect. 8.12 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 519-523, 2003.

Harry J. Smith, Table of n, a(n) for n = 1..20000
Jason Kimberley, Index of expansions of sqrt(d) in base b
R. J. Nemiroff & J. Bonnell, The first 1 million digits of the square root of 8
R. J. Nemiroff & J. Bonnell, Plouffe's Inverter, The first 1 million digits of the square root of 8
Ana Rechtman, Juin 2023, 3e défi, Images des Mathématiques, CNRS, 2023.
S. Uchiyama, On the distribution of integers representable as a sum of two h-th powers, J. Fac. Sci. Hokkaido Univ. Ser. I, 18, 124-127, 1964/1965.
Eric Weisstein's World of Mathematics, Moving Ladder Problem
Index entries for algebraic numbers, degree 2

Cf. A040005 (continued fraction).

Cf. A331250, A331251.

SetDefaultRealField(RealField(100)); Sqrt(8); // Vincenzo Librandi, Feb 13 2020

evalf(2^(3/2)) ; # R. J. Mathar, Jul 15 2013

RealDigits[N[Sqrt[8],200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011 *)

default(realprecision, 20080); x=sqrt(8); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010466.txt", n, " ", d));  \\ Harry J. Smith, Jun 02 2009

Equals 1 + Sum_{n>=1} ( Product_{k=1..n} (2k+1)/(4k) ). - Bruno Berselli, Mar 16 2014
Equals 2*A002193. - R. J. Mathar, Jan 14 2021
From Peter Bala, Mar 01 2022: (Start)
Equals 3*Sum_{n >= 0} (1/(4*n+1) - 1/(4*n-3))*binomial(1/2,n). Cf. A002580 and A175576.
Equals 4*hypergeom([-1/2, -3/4], [5/4], -1). (End)
Equals 8 * A020765. - R. J. Mathar, Aug 24 2023

A093341 Decimal expansion of "lemniscate case".

Original entry on oeis.org

1, 8, 5, 4, 0, 7, 4, 6, 7, 7, 3, 0, 1, 3, 7, 1, 9, 1, 8, 4, 3, 3, 8, 5, 0, 3, 4, 7, 1, 9, 5, 2, 6, 0, 0, 4, 6, 2, 1, 7, 5, 9, 8, 8, 2, 3, 5, 2, 1, 7, 6, 6, 9, 0, 5, 5, 8, 5, 9, 2, 8, 0, 4, 5, 0, 5, 6, 0, 2, 1, 7, 7, 6, 8, 3, 8, 1, 1, 9, 9, 7, 8, 3, 5, 7, 2, 7, 1, 8, 6, 1, 6, 5, 0, 3, 7, 1, 8, 9, 7, 2, 7, 7, 7, 7

Offset: 1

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Apr 26 2004

			1.854074677301371918433850347195260046217598823521766905585928045056021...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, 9th printing. New York: Dover, 1972, Section 18.14.7, p. 658.
Jonathan Borwein & Peter Borwein, A Dictionary of Real Numbers. Pacific Grove, California: Wadsworth & Brooks/Cole Advanced Books & Software, 1990, p. iii.
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 6.1 Gauss' Lemniscate Constant, pp. 421-422.

Harry J. Smith, Table of n, a(n) for n = 1..5000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Section 18.14.7, p. 658.
G. Mingari Scarpello and D. Ritelli, On computing some special values of hypergeometric functions, arXiv:1212.0251 [math.CA], 2012-2014, eq. (4.1).
Eric Weisstein's World of Mathematics, Lemniscate Case.
I. J. Zucker and G. S. Joyce, Special values of the hypergeometric series II, Math. Proc. Camb. Phil. Soc. 131 (2001) 309-319, (2.3).

Cf. A064853, A062539, A105372, A153396, A175576, A224268.

evalf( EllipticK(1/sqrt(2)) ); # R. J. Mathar, Aug 28 2013

RealDigits[ N[ Gamma[1/4]^2 / (4*Sqrt[Pi]), 105]][[1]] (* Jean-François Alcover, Oct 04 2011 *)
RealDigits[N[EllipticK[1/2], 105]][[1]] (* Vaclav Kotesovec, Feb 22 2015 *)

{ allocatemem(932245000); default(realprecision, 5080); x=gamma(1/4)^2/(4*(Pi)^(1/2)); for (n=1, 5000, d=floor(x); x=(x-d)*10; write("b093341.txt", n, " ", d)); } \\ Harry J. Smith, Jun 19 2009

Pi/agm(sqrt(2),2) \\ Charles R Greathouse IV, Feb 04 2015

ellK(1/sqrt(2)) \\ Charles R Greathouse IV, Feb 04 2025

GAMMA(1/4)^2/(4*(Pi)^(1/2)). - Pab Ter (pabrlos(AT)yahoo.com), May 24 2004
Also equals ellipticK(1/sqrt(2)) = Pi/2*hypergeom([1/2,1/2],[1],1/2),
or also the smallest positive root of cs(x/sqrt(2)|-1), where cs is the Jacobi elliptic function, or also the real half-period of the Weierstrass Pe function (Cf. Finch p. 422). - Jean-François Alcover, Apr 30 2013, updated Aug 01 2014
From Peter Bala, Feb 22 2015: (Start)
Equals Integral_{x = 0..oo} 1/sqrt(1 + x^4) dx = 2 * Integral_{x = 0..1} 1/sqrt(1 + x^4) dx = sqrt(2) * Integral_{x = 0..1} 1/sqrt(1 - x^4) dx.
Equals 2 * Sum {n >= 0} (-1/4)^n * binomial(2*n,n) * 1/(4*n + 1). (End)
Equals A062539 / sqrt(2). - Amiram Eldar, May 04 2022
Equals 1/A105372 = A175576/2 = 2*A224268. - Hugo Pfoertner, Aug 27 2024

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 24 2004

A290570 Decimal expansion of Integral_{0..Pi/2} dtheta/(cos(theta)^3 + sin(theta)^3)^(2/3).

Original entry on oeis.org

1, 7, 6, 6, 6, 3, 8, 7, 5, 0, 2, 8, 5, 4, 4, 9, 9, 5, 7, 3, 1, 3, 6, 8, 9, 4, 9, 9, 6, 4, 8, 4, 3, 8, 7, 0, 2, 5, 7, 1, 8, 6, 8, 5, 3, 8, 2, 0, 2, 5, 5, 7, 5, 3, 0, 1, 2, 6, 9, 0, 5, 2, 4, 1, 8, 3, 5, 4, 5, 3, 0, 0, 1, 7, 2, 8, 1, 0, 7, 9, 1, 3, 6, 0, 5, 4, 8, 6, 9, 9, 3, 3, 3, 3, 3, 8, 3, 5, 8, 7, 2, 1, 9, 3, 4

Offset: 1

Jean-François Alcover, Aug 07 2017

			1.766638750285449957313689499648438702571868538202557530126905241835453...

Oscar S. Adams, Elliptic Functions Applied to Conformal World Maps, Special Publication No. 112 of the U.S. Coast and Geodetic Survey, 1925. See constant K p. 9 and previous pages.

StackExchange, Question 2380131 Evaluate in closed form Integral_{0..Pi/2} dtheta/(cos(theta)^3+sin(theta)^3)^(2/3), August 2017.

Cf. A073005 (Gamma(1/3)), A073006 (Gamma(2/3)), A197374 (Beta(1/3,1/3)).

Cf. A104133, A104134, A197374.

RealDigits[(1/3)*Gamma[1/3]^2/Gamma[2/3], 10, 105]

(1/3)*gamma(1/3)^2/gamma(2/3) \\ Michel Marcus, Aug 07 2017

Equals (1/3)*Beta(1/3,1/3).
Equals (1/3)*Gamma(1/3)^2/Gamma(2/3).
Equals A197374/3. - Michel Marcus, Jun 08 2020
From Peter Bala, Mar 01 2022: (Start)
Equals 2*Sum_{n >= 0} (1/(3*n+1) + 1/(3*n-2))*binomial(1/3,n). Cf. A002580 and A175576.
Equals Sum_{n >= 0} (-1)^n*(1/(3*n+1) - 1/(3*n-2))*binomial(1/3,n).
Equals hypergeom([1/3, 2/3], [4/3], 1) = (3/2)*hypergeom([-1/3, -2/3], [4/3], 1) = 2*hypergeom([1/3, 2/3], [4/3], -1) = hypergeom([-1/3, -2/3, 5/6], [4/3, -1/6], -1). (End)

A175575 Decimal expansion of (Gamma(3/4))^2 / Pi^(3/2) .

Original entry on oeis.org

2, 6, 9, 6, 7, 6, 3, 0, 0, 5, 9, 4, 1, 8, 9, 6, 7, 8, 3, 3, 3, 9, 6, 7, 8, 6, 1, 1, 7, 7, 7, 7, 6, 3, 6, 6, 3, 8, 2, 9, 3, 4, 4, 8, 2, 7, 2, 1, 5, 2, 0, 0, 6, 5, 1, 6, 9, 9, 7, 3, 3, 1, 5, 9, 3, 1, 9, 4, 1, 4, 9, 4, 2, 4, 3, 2, 5, 7, 8, 4, 1, 4, 0, 7, 7, 9, 6, 0, 6, 8, 6, 1, 3, 7, 6, 6, 8, 8, 5, 7, 3, 6, 2, 8, 2

Offset: 0

R. J. Mathar, Jul 15 2010

Entry 34 d of chapter 11 of Ramanujan's second notebook.

			0.2696763005941896783339678...

Bruce C. Berndt, Chapter 11 of Ramanujan's second notebook, Bull. Lond. Math. Soc. vol 15 no 4 (1983) 273-320.

Maple
```
GAMMA(3/4)^2/Pi^(3/2) ; evalf(%) ;
```

RealDigits[Gamma[3/4]^2/Pi^(3/2),10,120][[1]] (* Harvey P. Dale, Mar 16 2021 *)

Equals A068465^2 / (A000796 * A002161 ) = 1/A175576.

Equals (5/16)*hypergeom([1/4, -3/4], [3/2], 1). - Peter Bala, Mar 02 2022

A186642 Decimal expansion of the "squircle" perimeter.

Original entry on oeis.org

7, 0, 1, 7, 6, 9, 7, 9, 4, 3, 5, 6, 4, 0, 4, 1, 6, 4, 7, 1, 0, 6, 4, 9, 4, 1, 6, 3, 9, 3, 1, 8, 1, 1, 6, 9, 3, 9, 8, 0, 0, 8, 7, 5, 0, 4, 9, 7, 2, 4, 4, 9, 3, 4, 3, 2, 2, 8, 8, 6, 1, 0, 3, 5, 6, 0, 7, 3, 9, 2, 2, 1, 1, 6, 1, 8, 1, 8, 8, 8, 3, 5, 1, 3, 2, 3, 8, 8, 3, 9, 3, 0, 0, 5, 0, 3, 4, 0, 7, 1

Offset: 1

Jean-François Alcover, Feb 25 2011

This squircle constant can also be computed as a series in terms of incomplete beta function with coefficients from sequences A002596 and A120777:
  a(n) = (-1)^(n+1) numerator((2n-3)!!/n!) ( sequence A002596);
  b(n) = denominator(binomial(2n+2, n+1)/2^(2n+1)) ( sequence A120777).
  Generic term:
  u(n) = (a(n)/b(n-1))*beta(1/2, (6n+1)/4, 1-(3/2)*n).
  Here is the series computed up to 5 terms:
  4*2^(3/4) + sum(u(n), {n, 1, 5}) =
  4*2^(3/4) + beta(1/2, 7/4, -1/2) - (1/4)*beta(1/2, 13/4, -2) + (1/8)* beta(1/2, 19/4, -7/2) - (5/64)*beta(1/2, 25/4, -5) + (7/128)*beta(1/2, 31/4, -13/2).
  It evaluates to 7.018901897260651...
  Numeric check with 10000 terms:
  4*2^(3/4) + sum(u(n), {n, 1, 10000}) = 7.017697943556135...

			7.01769794356404...

Eric Weisstein's World of Mathematics, Squircle

Cf. A175576 (unit squircle area).

First @ RealDigits[N[2*Integrate[Sqrt[1 + x^(3/2)/(1 - x)^(3/2)]/x^(3/4), {x, 0, 1/2}], 100]]
(* This other series formula gives 100 correct digits: *)
First @ RealDigits[1/Sqrt[Pi]*NSum[(-1)^(n+1)*Gamma[n - 1/2]*Beta[1/2, (6n + 1)/4, 1 - (3/2)n] / n!, {n, 0, Infinity},WorkingPrecision -> 100, Method -> "AlternatingSigns"], 10, 100]

-((3^(1/4) MeijerG[{{1/3, 2/3, 5/6, 1, 4/3}, {}}, {{1/12, 5/12, 7/12, 3/4, 13/12}, {}}, 1])/(16 Sqrt[2] Pi^(7/2) Gamma[5/4])). - Eric W. Weisstein, Oct 25 2011

A347430 Simple continued fraction expansion of Pi^(3/2)/Gamma(3/4)^2.

Original entry on oeis.org

3, 1, 2, 2, 2, 1, 8, 1, 2, 1, 2, 9, 8, 6, 56, 5, 38, 1, 2, 1, 5, 1, 5, 1, 2, 10, 3, 10, 741, 1, 5, 3, 3, 1, 5, 2, 3, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 7, 2, 3, 3, 4, 4, 1, 11, 1, 2, 1, 1, 1, 1, 1, 5, 1, 64, 1, 1, 2, 7, 1, 5, 98, 2, 2, 2, 1, 1, 1, 1, 1, 5, 1, 3, 1

Offset: 0

Adam Filinovich, Sep 01 2021

			3+1/(1+1/(2+1/(2+1/(2+...)))).

M. Parker, What is the area of a Squircle?, Youtube video (2021).
Eric Weisstein's World of Mathematics, Squircle

Cf. A175576 for decimal expansion.

convert(8*GAMMA(5/4)^2/sqrt(Pi), confrac, 84); # Peter Luschny, Sep 02 2021

ContinuedFraction[Pi^(3/2)/Gamma[3/4]^2, 84] (* Michael De Vlieger, Sep 01 2021 *)

contfrac(Pi^1.5/gamma(3/4)^2) \\ Michel Marcus, Sep 02 2021

Equals sqrt(2)*Pi/agm(1,sqrt(2)) (arithmetic-geometric mean).

Equals 8*Gamma(5/4)^2/sqrt(Pi). - Peter Luschny, Sep 02 2021

A384563 Decimal expansion of Beta(1/4,1/4).

Original entry on oeis.org

7, 4, 1, 6, 2, 9, 8, 7, 0, 9, 2, 0, 5, 4, 8, 7, 6, 7, 3, 7, 3, 5, 4, 0, 1, 3, 8, 8, 7, 8, 1, 0, 4, 0, 1, 8, 4, 8, 7, 0, 3, 9, 5, 2, 9, 4, 0, 8, 7, 0, 6, 7, 6, 2, 2, 3, 4, 3, 7, 1, 2, 1, 8, 0, 2, 2, 4, 0, 8, 7, 1, 0, 7, 3, 5, 2, 4, 7, 9, 9, 1, 3, 4, 2, 9, 0, 8, 7, 4, 4, 6, 6, 0, 1, 4, 8, 7, 5, 8, 9

Offset: 1

Stefano Spezia, Jun 03 2025

			7.416298709205487673735401388781040184870395294...

Michael I. Shamos, A catalog of the real numbers, (2007). See p. 406.

Similar constants Beta(1/k,1/k): A000796 (k=2), A197374 (k=3).

Cf. A002161, A068466, A175576, A377731.

Mathematica
```
RealDigits[Beta[1/4,1/4],10,100][[1]]
```

Equals Gamma(1/4)^2/sqrt(Pi) = A068466^2/A002161.

Equals 2*A175576 = 3*A377731. - Hugo Pfoertner, Jun 03 2025

cp's OEIS Frontend

A010466 Decimal expansion of square root of 8.

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A093341 Decimal expansion of "lemniscate case".

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A290570 Decimal expansion of Integral_{0..Pi/2} dtheta/(cos(theta)^3 + sin(theta)^3)^(2/3).

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A175575 Decimal expansion of (Gamma(3/4))^2 / Pi^(3/2) .

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A186642 Decimal expansion of the "squircle" perimeter.

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A347430 Simple continued fraction expansion of Pi^(3/2)/Gamma(3/4)^2.

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A384563 Decimal expansion of Beta(1/4,1/4).

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